EXPLORING THE TOOLKIT OF JEAN BOURGAIN

Gian-Carlo Rota asserted in his article, "Ten lessons I wish I had been taught" [Notices of the American Mathematical Society 44 (1997), no. 1, 22-25], that "every mathematician only has a few tricks". The sheer breadth and ingenuity in the work of Jean Bourgain may at first glance appear to be a counterexample to this maxim. However, as we hope to illustrate in this article, even Bourgain relied frequently on a core set of tools, which formed the base from which problems in many disparate mathematical fields could then be attacked. We discuss a selected number of these tools here, and then perform a case study of how an argument in one of Bourgain's papers can be interpreted as a sequential application of several of these tools.